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%I #7 Dec 06 2024 11:10:18
%S 49,125,243,343,1681,1849,3249,4913,6724,6859,8649,9801,11449,13689,
%T 13924,17576,20449,24389,24649,28125,28224,29791,31212,36125,37249,
%U 40328,42849,45125,57121,59049,63001,66049,68921,79507,83349,85849,94249,99127,106929,110224
%N Powerful numbers k such that both k-1 and k+1 are in A126706.
%C Contains certain powerful k in A246547 (perfect powers of primes) or in A286708 (powerful numbers that are not prime powers).
%C Contains certain Achilles numbers (in A052486); a(20) = 28125 = 3^2 * 5^5.
%H Michael De Vlieger, <a href="/A378629/b378629.txt">Table of n, a(n) for n = 1..10000</a>
%e Let S = A126706, the sequence of k that are neither squarefree nor prime powers.
%e {1, 4, 8, 9} are not in the sequence since S(1) = 12.
%e a(1) = 49 = 7^2 since both 48 = 2^3 * 3 and 50 = 2 * 5^2 are in S.
%e 64 is not in the sequence since 65 is squarefree.
%e a(2) = 125 = 5^3 since both 124 = 2^2 * 41 and 126 = 2 * 3^2 * 7 are in S.
%e 128 is not in the sequence since 127 is prime.
%e a(3) = 243 = 3^5 since both 242 = 2 * 11^2 and 244 = 2^2 * 61 are in S.
%e a(7) = 3249 = 3^2 * 19^2, since both 3248 = 2^4 * 7 * 29 and 3250 = 2 * 5^3 * 13 are in S, etc.
%t With[{nn = 2^30}, Select[Union@ Flatten@ Table[a^2*b^3, {b, Surd[nn, 3]}, {a, Sqrt[nn/b^3]}], AllTrue[# + {-1, 1}, Nor[SquareFreeQ[#], PrimePowerQ[#] ] &] &] ]
%Y Cf. A001694, A013929, A024619, A052486, A126706, A246547, A286708.
%K nonn,easy,new
%O 1,1
%A _Michael De Vlieger_, Dec 03 2024